Single-molecule Forster resonance energy transfer (FRET) between fluorescent donor and acceptor labels attached to a protein or nucleic acid can be used to probe a molecules structure, dynamics and function. In these experiments, a molecule is either immobilized on a surface or diffuses through a spot illuminated by a laser, and the donor is excited. The donor can emit a photon or transfer the excitation to an acceptor which then can emit a photon of a different color. The rate of transfer depends on (interdye distance)-6 and this is why there is information about conformational dynamics (FRET is the optical analog of the NOE in NMR that is used in structure determination). The output of these experiments is a photon trajectory (the color of the photons emitted by the donor differ from those emitted by the acceptor). The observed sequence of photons can be binned, and a histogram of the FRET efficiencies for each bin, defined as the fraction of the photons emitted from the acceptor, can be constructed. The shape of the histogram depends on the conformational states of the molecule and their interconversion rates. When a single molecule is excited by a train of laser pulses, it is not only possible to detect the colors and arrival times of the emitted photons, but also the time interval between the laser pulse and the photon. This so-called delay time is related to the fluorescence lifetime of the fluorophore. The fluorescence lifetime depends on the rate of energy transfer and hence decreases as the donor and acceptor come closer together. During the last reporting period, we have generalized our previous work on FRET efficiency histograms to include delay times. Our main theoretical contribution was to derive an exact expression for the joint distribution of the numbers of donor and acceptor photons and donor lifetimes in a bin that treats the influence of conformational dynamics on all time scales. Perhaps the most interesting finding is that the connectivity of the underlying conformational states can be determined directly by simple visual inspection of the projection of the experimental joint distribution on the efficiency-lifetime plane. This year we have been working on a more sophisticated approach, that does not require binning, in which all the information available in a photon trajectory is used to extract a model of conformation dynamics. The basic idea to construct the likely-hood that the color, arrival and delay time of each and every photon that is observed is consistent with a proposed model of the dynamics. This likely-hood is then maximized with respect to the model parameters. We are now investigating to what extent the extra information contained in the lifetimes increases the reliability and accuracy of conclusions about structure and dynamics that can be obtained from the analysis of experiments. Thermodynamically independent receptors on the surface of a cell are in fact coupled due to the finite rate of ligand diffusion. Because the binding sites compete for ligands, the rate of binding to one receptor depends on the occupancy of the other receptors. This was first pointed out by Berg and Purcell in their classic paper on chemotaxis. Moreover, they showed that such diffusion-induced interactions had a surprising consequence. Specifically, diffusion places a physical limit on how accurately a cell can determine the concentration of an attractant in its surrounding environment. In this reporting period, we have developed a theory for the role of diffusion in the kinetics of reversible ligand binding to receptors on a cell surface or to a macromolecule with multiple binding sites. A formalism was based on a Markovian master equation for the distribution function of the number of occupied receptors containing rate constants that depend on the ligand diffusivity, and was used to derive (1) a nonlinear rate equation for the mean number of occupied receptors and (2) an analytical expression for the relaxation time that characterizes the decay of equilibrium fluctuations of the occupancy of the receptors. The relaxation time was shown to depend on the ligand diffusivity and concentration, the number of receptors, the cell radius, and intrinsic association/dissociation rate constants. This result was then used to estimate the accuracy of the ligand concentration measurements by the cell, which,according to the Berg-Purcell model, is related to fluctuations in the receptor occupancy, averaged over a finite interval of time. Specifically, a simple expression (which is exact in the framework of our formalism) was derived for the variance in the measured ligand concentration in the limit of long averaging times. The dynamics of protein folding is often described as diffusion along a collective reaction coordinate in the presence of the corresponding potential of mean force. This is a tremendous simplification for such a complex precess but it has proven to be a remarkably useful framework for the analysis and interpretation of experimental data. In general, multidimensional dynamics can be reduced to the dynamics of a single coordinate only when the dynamics along all other coordinates is sufficiently fast to establish local equilibrium. We asked the question if such a coordinate can exist under less restrictive conditions and discovered that the splitting probabilities of states in the region of configuration space that separates reactants and products have a remarkable property. Assuming that the splitting probability changes more slowly than any other coordinate, one can always project multidimensional diffusive dynamics onto it. The resulting one-dimensional diffusion equation is unfortunately not exact because the assumed separation of time scales does not hold in general. Nevertheless, we found that this equation has the surprising property that it always predicts the exact value of the number of transitions between reactants and products per unit time at equilibrium and hence the exact reaction rate. Thus in this precise sense, it is the perfect reaction coordinate. In the special case of two deep basins separated by a harmonic saddle, this equation is equivalent to the one that describes diffusion along a coordinate perpendicular to the transition state, defined as the surface starting from which reactants and products are reached with equal probability.